Dimensionless Problem Research Articles

Переход к безразмерной задаче в модели ветровой циркуляции

Переход к безразмерной задаче в модели ветровой циркуляции

Self-Diffusiophoresis of Slender Catalytic Colloids.

We consider the self-diffusiophoresis of axisymmetric particles using a continuum description where the interfacial chemical reaction is modeled by first-order kinetics with a prescribed axisymmetric distribution of rate-constant magnitude. We employ the standard macroscale framework where the interaction of solute molecules with the particle boundary is represented by diffusio-osmotic slip. The dimensionless problem governing the solute transport involves two parameters (the particle slenderness ϵ and the Damköhler number Da) as well as two arbitrary functions which describe the axial distributions of the particle shape and rate-constant magnitude. The resulting particle speed is determined throughout the solution of the accompanying problem governing the flow about the force-free particle. Motivated by experimental configurations, we employ slender-body theory to investigate the asymptotic limit ϵ ≪ 1. In doing so, we seek algebraically accurate approximations where the asymptotic error is smaller than a positive power of ϵ. The resulting approximations are thus significantly more useful than those obtained in the conventional manner, where the asymptotic expansion is carried out in inverse powers of ln ϵ. The price for that utility is that two linear integral equations need to be solved: one governing the axial solute-sink distribution and the other governing the axial distribution of Stokeslets. When restricting the analysis to spheroidal particles, no need arises to solve for the Stokeslet distribution. The integral equation governing the solute-sink distribution is then solved using a numerical finite-difference scheme. This solution is supplemented by a large-Da asymptotic analysis, wherein a subtle nonuniformity necessitates a careful treatment of the regions near the particle ends. The simple approximations thereby obtained are in excellent agreement with the numerical solution.

Dissipative flow of hybrid nanoliquid (H2O-aluminum alloy nanoparticles) with thermal radiation

An improvement in heat transfer fluid is one of the important tasks which can be achieved through implementation of hybrid nanoparticles. Hence the aim of this study is to communicate flow of hybrid nanoliquid by stretchable rotating disk. The considered hybrid nanoliquid is a combination of AA7072 and AA7075 nanoparticles and water. Law of energy incorporates the thermal radiation, dissipation and irregular heat source terms. Von Karman variables lead to the dimensionless problem under consideration. Reduced system is tackled numerically via NDSolve based Shooting scheme. Graphical descriptions are made against various physical variables of interest. Moreover rate of heat transfer at disk is estimated. Present work pointed out that local heat transfer rate accelerates for higher thermal radiation and thermal slip parameters whereas decaying trend is seen for higher estimations of Eckert number and (ESHS) parameter. It is noted that fluid velocity reduces for higher volumetric fraction. Furthermore hybrid nanomaterials have great influence throughout our analysis when compared with regular nanomaterials. The peculiar outcomes achieved in this study are important in industrial research, academics and discussion about heat transfer analysis in flow of two types of nanoliquids by keeping the regular fluid fixed. Finally key points of present analysis are included.

MHD peristaltic motion of Johnson–Segalman fluid in an inclined channel subject to radiative flux and convective boundary conditions

Background: In abundant of a digestive tract like smooth muscle tissue, human gastrointestinal tract contracts in sequence to generate a peristaltic wave, which pushes a food along the tract. The peristaltic motion contains circular relaxation smooth muscles, then their shrinkage (contraction) behind the chewed material to keep it from moving backward, then longitudinal contraction to shove it ahead. Therefore, we have conducted a theoretical investigation on peristaltic transport in flow of Johnson-Segalman liquid subject to inclined magnetic field. The energy equation is developed with extra heat transport assumptions like thermal radiative flux and dissipation. The channel walls are heated convectively.Methods: Dimensionless problems subject to small Reynolds number and long wavelength are tackled. Perturbation technique is implemented for small Weissenberg number.Results: The physical importance of involved parameters that directly affect the heat transfer rate temperature and velocity. The pertinent variables are amplitude ratio, wave number, Reynolds number, Hartman number, Prandtl number, Weissenberg number, thermal radiative heat flux, Biot number, elasticity variables and Froude number are graphically discussed. The obtained outcome shows that the velocity field increases against higher values of elasticity variables but velocity the material decays through higher fluid parameter. Temperature field declines through higher Hartman number. Furthermore, it is also examined that the heat transfer rate decays against rising Hartman number.Conclusions: The impact of complaint walls on radiative peristaltic transport of Johnson–Segalman liquid in symmetric channel subject to inclined angle. The influence of Johnson–Segalman variable on the velocity field shows decreasing behavior. Velocity also declines against larger Hartman number. Temperature and heat transfer rate boosts through rising values of E1 E2 while decays versus larger E3. Furthermore, reduction in heat transfer coefficient is observed when the values of α and Br are increased.

Analysis of activation energy and melting heat transfer in MHD flow with chemical reaction

This paper concentrates on aspects of melting heat transfer along with binary chemical reaction and activation energy in stagnation flow over a sheet of variable thickness saturated in a porous medium. Radiative and magnetohydrodynamic effects are taken into account. A more innovative surface condition (i.e. melting heat transfer condition) is utilized to explore the characteristics of heat transfer. By making use of adequate transformations, a nonlinear system of ordinary differential equations is achieved. The homotopic approach is used to obtain the series solution of the considered dimensionless problem. Graphical analysis of various variables is given for velocity, thermal and concentration distributions. A decreasing behavior of the concentration field is seen for the destructive chemical reaction parameter. The concentration field gets intensified for larger values of the activation energy parameter and the velocity field decreases for larger porosity parameter.

Entropy generation analysis for a thin couple stress film flow over an inclined surface with Newtonian cooling

In this article, a computational approach to monitor entropy production in a fully developed gravity-driven couple stress film flow on an inclined heated surface subjected to convective cooling is examined. The free end of the thin film is maintained in a manner that follows the Newtonian law of cooling. Exact solutions of the dimensionless boundary-value problems (BVP) are obtained and utilized for the entropy expression and the heat irreversibility ratio along the heated plate. The effects of pertinent parameters on the flow and thermal build up are shown graphically and explained based on physical laws. Most importantly, the rate of heat transfer is shown to be a decreasing function of the couple stress inverse while the skin friction remains constant along the heated wall. The present computation is relevant in coating and drying processes and several metallurgical engineering.

Eccentricity Effects on Bearing Capacity of Strip and Square Footings on Two-Layered Clay

The undrained ultimate capacity to combined (M, V) loading of strip and square foundations on a two-layered clay deposit is investigated. Extensive two and threedimensional parametric finite element analyses are carried out in terms of the ratio of the undrained shear strength of two layers, the normalized thickness of the top layer and the normalized load eccentricity. Two basic cases are separately investigated, referring to strength ratios higher or lower than unity. The assumption of effective width of the foundations, to account for eccentricity, generally adopted for homogeneous soil, is extended for two layers. The results are mainly presented in terms of modified bearing capacity factors for strip or square footings for a wide range of dimensionless problem parameters. However, for a better visual understanding of how the bending moment affects the ultimate vertical load failure loci diagrams (“yield surfaces”) in M, V space are also presented. Emphasis is given to developing insight into the particular failure mechanisms of most examined cases.

Thermally radiated squeezed flow of magneto-nanofluid between two parallel disks with chemical reaction

This research is exhibited to visualize the squeezed flow of magneto-nanoliquid between two parallel disks. Transportations of heat and mass are characterized through thermal radiation and chemical reaction. Suitable similarity variables lead to dimensionless problem. The homotopic technique is adopted to find out the solutions. Behavior of sundry variables is declared graphically. Physical quantities of curiosity such as skin friction and Nusselt number at both disks are estimated and elaborated. Our investigation depicts that thermal field is augmented via radiation and Brownian diffusion variables. Besides, comparative table is also designed to validate our present outcomes with previous limiting study.

Boundary layer flow of MHD generalized Maxwell fluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary

Abstract The aim of this article is to investigate the unsteady natural convection flow of Maxwell fluid with fractional derivative over an exponentially accelerated infinite vertical plate. Moreover, slip condition, radiation, MHD and Newtonian heating effects are also considered. A modern definition of fractional derivative operator recently introduced by Caputo and Fabrizio has been used to formulate the fractional model. Semi analytical solutions of the dimensionless problem are obtained by employing Stehfest’s and Tzou’s algorithms in order to find the inverse Laplace transforms for temperature and velocity fields. Temperature and rate of heat transfer for non-integer and integer order derivatives are computed and reduced to some known solutions from the literature. Finally, in order to get insight of the physical significance of the considered problem regarding velocity and Nusselt number, some graphical illustrations are made using Mathcad software. As a result, in comparison between Maxwell and viscous fluid (fractional and ordinary) we found that viscous (fractional and ordinary) fluids are swiftest than Maxwell (fractional and ordinary) fluids.

Multiple slips effects on MHD SA-Al2O3 and SA-Cu non-Newtonian nanofluids flow over a stretching cylinder in porous medium with radiation and chemical reaction

The purpose of this communication is to examine the collective influence of velocity, and thermal slips on magnetohydrodyanmics (MHD) SA-Al2O3 and SA-Cu non-Newtonian nanofluids flow over a stretching cylinder in porous medium together with thermal radiation and chemical reaction effects. Sodium Alginate (SA-NaAlg) is taken as non-Newtonian base fluid. Two types of nanoparticles alumina or aluminum oxide (Al2O3) and copper (Cu) are suspended in sodium alginate (SA) which is taken as base fluid, an example of non-Newtonian Casson fluid. The formulated nonlinear partial differential equations with auxiliary boundary conditions are transformed into non-dimensional form by applying suitable similarity transformations. The resulting dimensionless problem is solved numerically using shooting and fourth order Runge-Kutta method. The impacts of various thermophysical parameters on local skin-friction, local Nusselt number, temperature and velocity are analyzed through graphs as well as in tabular form and discussed in detail. A comparison between SA-Al2O3 and SA-Cu nanofluids is clearly shown and in limiting sense the present results are compared with published results from literature. The results show that with magnetic parameter, skin-friction and Nusselt number both decreased, and Nusselt numbers are the highest in case of Al2O3 than Cu nanoparticles.

Impact of compliant walls on magnetohydrodynamics peristalsis of Jeffrey material in a curved configuration

Primary aim of current attempt is to analyze the peristaltic flow of non-Newtonian material in a curved channel subject to two salient features namely the Soret and Dufour and radial magnetic field. Channel walls are of compliant characteristics. Problem formulations for constitutive equations of Jeffrey fluid are made. Lubrication approach is implemented for the simplification of mathematical analysis. Dimensionless problems of stream function, temperature and concentration are computed numerically. Characteristics of distinct variables on the velocity, temperature, coefficient of heat transfer and concentration are examined. Besides this graphical results indicates that velocity profile enhances significantly for compliant wall parameters however due to the resistance characteristics of Lorentz force velocity profile decays. Furthermore it is noted that temperature profile enhances for larger Dufour number however reverse behavior is noticed in the concentration profile when Soret and Schmidt numbers are increased.

Numerical Simulations with the Galerkin Least Squares Finite Element Method for the Burgers' Equation on the Real Line

In this work we present an efficient Galerkin least squares finite element scheme to simulate the Burgers' equation on the whole real line and subjected to initial conditions with compact support. The numerical simulations are performed by considering a sequence of auxiliary spatially dimensionless Dirichlet's problems parameterized by its numerical support $\tilde{K}$. Gaining advantage from the well-known convective-diffusive effects of the Burgers' equation, computations start by choosing $\tilde{K}$ so it contains the support of the initial condition and, as solution diffuses out, $\tilde{K}$ is increased appropriately. By direct comparisons between numerical and analytic solutions and its asymptotic behavior, we conclude that the proposed scheme is accurate even for large times, and it can be applied to numerically investigate properties of this and similar equations on unbounded domains.

Simultaneous effects of melting heat transfer and inclined magnetic field flow of tangent hyperbolic fluid over a nonlinear stretching surface with homogeneous–heterogeneous reactions

The purpose of present article is to investigate homogeneous–hetrogeneous reactions and melting heat transfer in the flow of magnetohydrodynamic (MHD) mixed convective flow of hyperbolic tangent fluid. Flow by nonlinear stretching sheet is addressed in presence of non-uniform heat source/sink. Governing equations through conservations laws are obtained. Series solutions of dimensionless problems are developed within the frame of homotopic theory. Convergence solution is achieved and suitable values are found. The velocity, temperature and concentration are analyzed graphically within the frame of various pertinent variables. Skin friction coefficient, local Nusselt number and wall concentration have been scrutinized through plots. An increase in space and temperature dependent non-uniform heat source/sink variables show rise to temperature.

Electromagneto squeezing rotational flow of Carbon (C)-Water (H2O) kerosene oil nanofluid past a Riga plate: A numerical study.

This article predicts the electromagneto squeezing rotational flow of carbon-water nanofluid between two stretchable Riga plates. Riga plate is known as electromagnetic actuator which is the combination of permanent magnets and a span wise aligned array of alternating electrodes mounted on a plane surface. Mathematical model is developed for the flow problem with the phenomena of melting heat transfer, viscous dissipation and heat generation/absorption. Water and kerosene oil are utilized as the base fluids whereas single and multi-wall carbon nanotubes as the nanomaterials. Numerical solutions of the dimensionless problems are constructed by using built in shooting method. The correlation expressions for Nusselt number and skin friction coefficient are developed and examined through numerical data. Characteristics of numerous relevant parameters on the dimensionless temperature and velocity are sketched and discussed. Horizontal velocity is found to enhance for higher modified Hartman number.

Impact of thermal radiation on electrical MHD flow of nanofluid over nonlinear stretching sheet with variable thickness

The present paper addresses magnetohydrodynamics (MHD) flow of nanofluid towards nonlinear stretched surface with variable thickness in the presence of electric field. The analysis is presented with viscous dissipation, Joule heating, and chemical reaction. Characteristics of heat transfer are analyzed with the electric field and variable thickness phenomenon. The partial differential equations are converted into dimensionless ordinary differential equations by employing suitable transformations. Implicit finite difference scheme is implemented to solve the governing dimensionless problems. Behaviors of several sundry variables on the flow and heat transfer are scrutinized. Skin friction coefficient, the local Nusselt number local Sherwood number are presented and evaluated. It is observed that the skin friction, the rate of heat and mass transfer reduces with a rise in wall thickness. Electric field enhances the nanofluid velocity and temperature but reduced the concentration. Thermal radiation is sensitive to an increase in the nanofluid temperature and thicker thermal boundary layer thickness. Obtained results are also compared with the available data in the limiting case and good agreement is noted.

Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo-Fabrizio derivatives

This article is focused on heat transfer analysis in the unsteady flow of a generalized Maxwell fluid over an oscillating vertical flat plate with constant temperature. The well-known equation of the Maxwell fluid with classical derivatives, describing the unidirectional and one-dimensional flow, has been generalized to a non-integer-order derivative, known as fractional derivative, with free convection term of buoyancy. A new definition of the fractional derivative introduced by Caputo and Fabrizio has been used in the mathematical formulation of the problem. Exact solution of the dimensionless problem has been obtained by using the Laplace transform. These solutions are expressed with complementary error and modified Bessel functions. Similar solutions for classical Maxwell and Newtonian fluids and generalized Newtonian fluid performing the same motion are obtained as limiting cases of our general results. Graphical illustrations show that the velocity profiles corresponding to a generalized Maxwell fluid are similar to those for an ordinary Maxwell fluid when the fraction order approaches 1. A comparison amongst four different types of fluids is also shown graphically.

Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives

This article is focused on natural convection of unsteady flow of generalized Maxwell fluid over an oscillating vertical flat plate with constant temperature at the boundary. The Maxwell fluid with classical derivatives, describing one dimensional flow has been generalized to non-integer order derivatives known as fractional derivative with term of buoyancy. A modern definition of fractional derivative, recently introduced by Caputo and Fabrizio has been used to formulate the considered problem. Semi analytical solutions of the dimensionless problem have been obtained by using the Laplace transform. The solutions for temperature, velocity and shear stress are obtained with numerical inversion techniques of Laplace transform namely, Stehfest’s and Tzou’s algorithms. At the end, graphical illustrations for temperature, velocity, Nusselt number and shear stress are plotted. We have studied especially the influence of fractional parameter on temperature, velocity and shear stress respectively. We have observed that temperature can be enhanced for increasing the fractional parameter α while velocity and shear stress can be increased by decreasing the value of fractional parameter α.

Impact of melting heat transfer and nonlinear radiative heat flux mechanisms for the generalized Burgers fluids

The present paper deals with the analysis of melting heat and mass transfer characteristics in the stagnation point flow of an incompressible generalized Burgers fluid over a stretching sheet in the presence of non-linear radiative heat flux. A uniform magnetic field is applied normal to the flow direction. The governing equations in dimensional form are reduced to a system of dimensionless expressions by implementation of suitable similarity transformations. The resulting dimensionless problem governing the generalized Burgers is solved analytically by using the homotopy analysis method (HAM). The effects of different flow parameters like the ratio parameter, magnetic parameter, Prandtl number, melting parameter, radiation parameter, temperature ratio parameter and Schmidt number on the velocity, heat and mass transfer characteristics are computed and presented graphically. Moreover, useful discussions in detail are carried out with the help of plotted graphs and tables.

Effect logs of double diffusion on MHD Prandtl nano fluid adjacent to stretching surface by way of numerical approach

The current communication is carried to contemplate the unique and novel characteristics of nanofluids by constructing formulation of Prandtl fluid model. The fascinating aspects of thermo diffusion effects are also accounted in this communication. Mathematical modelling is performed by employing boundary layer approach. Afterwards, similarity variables are selected to convert dimensional non-linear system into dimensionless expressions. The solution of governing dimensionless problem is executed by shooting method (SM). Graphical evaluation is displayed to depict the intrinsic behavior of embedded parameters on dimensionless velocity, temperature, solutal concentration and nanoparticle concentration profiles. Furthermore, the numerical variation for skin friction coefficient, local Nusselt number, Sherwood number and nano Sherwood number is scrutinized through tables. The assurance of current analysis is affirmed by developing comparison with previous findings available in literature, which sets a benchmark for implementation of computational approach. It is inferred from the computation that concentration profile increases whereas Sherwood number decreases for progressive values of Dufour solutal number.

Mixed Convection Stagnation-Point Flow of Powell-Eyring Fluid with Newtonian Heating, Thermal Radiation, and Heat Generation/Absorption

AbstractThis article explores the steady mixed convection stagnation-point flow of Powell-Eyring fluid in the presence of thermal radiation, heat generation/absorption, and chemical reaction. Suitable transformations are considered to reduce nonlinear boundary layer partial differential equations into ordinary differential equations. Convergent series solutions of the resulting dimensionless problems are obtained. The effects of emerging physical parameters on the velocity, temperature, and concentration profiles are examined. Numerical values of skin-friction coefficient and local Nusselt and Sherwood numbers are tabulated and analyzed for different values of emerging parameters. A comparative analysis with existing solutions in a limiting sense is made.